Notes from April 3 Tuesday

Transcription

1 Notes from April 3 Tuesday With many weeks of ce108 covering higher level computing using matlab, it is time to go lower level so we could understand more of the computer terms used frequently in the computer field. One important area to cover is how data is stored in the computer memory. The basic types of information to be stored are integers, floating point numbers, characters (for word processing), memory addresses, and machine instructions. Matlab treats most numerical data as floating numbers, e.g., 1/3 is , but pretty much all other languages treat that as an integer operation, 1/3 yields a result of 0 for C, Fortran and C++. Integers are important as indexes for multi-dimensional arrays. But a good understanding of how a floating number is stored will also help in basic programming. Matlab uses double precision floating point numbers as default; that means every variable is store in 64 bits of memory. The Fortran default is single precision or 32-bit storage. 64-bit numbers have 15 digits of accuracy and 32-bit numbers have 7 digits of accuracy. In most cases, 15 digits of accuracy is more than enough, perhaps an overkill for many simple calculations. But the example below shows how cancellation could results when small differences of large numbers are accumulated. Consider the case of the summation of a simple cosine series with x and N given before the execution: term=1; sum=term; for i=2:2:n term=-term*x^2/(i*(i-1)); sum=sum+term; fprintf('%30.16f%30.16f\n',term,sum); end For x=1 and N=20, the results are excellent and the series converged readily to matlab answer for cos(1)= :

2 For x=20 and N=80, the cancellation effects of the large early terms caused many important digits of accuracy to be lost. For example, the term appears to be very accuracy, but after the first 15 digits, , the rest is garbage As shown above, the series will converge because the strength of N factorial will overcome x^n. But the damage was done early, so the series would not converge to cos(20)= no matter how many terms are used. A different algorithm must be used to achieve the results for large arguments. For discussion purposes, there is a need to understand how numbers are written in our everyday arithmetic exercises. We use base-10 for almost everything because we are born with 10 fingers. Base-10 has 10 literals: 0,1,2,3,4,5,6,7,8,9 and there is no literal for ten, instead, it is written as a combination of 2 literals, a 1 followed by a 0.

3 Mathematically, 10 is actually 1xB^1+0xB^0, in which B is the base of the number (B=ten for base-ten). Another example: 365 is actually 3xB^2+6xB^1+5xB^0. With this rule, all the numbers of other bases could be written. For computer application, binary-based numbers are important. One BIT (meaning one Binary digit) is the unit. Base-eight (Octal) is frequently used and that is represented by three bits. Hexadecimal, or basesixteen, is important and that is represented by four bits. The conversion table below shows the Octal and Hexadecimal literals in binary form: Octal Conversion Hexadecimal Conversion 0=000 0=0000 8=1000 1=001 1=0001 9=1001 2=010 2=0010 A=1010 3=011 3=0011 B=1011 4=100 4=0100 C=1100 5=101 5=0101 D=1101 6=110 6=0110 E=1110 7=111 7=0111 F=1111 Another unit that is important to computing is a BYTE. One byte is equal to 8 bits. One byte can store two hexadecimal digits. The 64-bit double precision floating point number can be stored in 8 bytes. Many documents on the computer are commonly measured in kilobytes or megabytes. The most famous powers of 2 are: 2^10=1024=1K. 2^20= =1M. 2^30= =1G. 2^40= =1T. Those values are approximately one thousand (1 Kilobyte), one million (1 Megabyte), one billion (1 Gigabyte) and one trillion (1 Terabyte). While you are scanning a document, you will notice settings such as 24- bit color, 48-bit color, 8-bit grayscale, 16-bit grayscale, Black and White, etc. 8-bit means the gradation of gray can be from to , 256 levels of gray, from 0 to 255 base-ten. 24-bit (24=3*8) color means the three basic colors RGB (Red, Green, Blue) each has 256 levels. For this reason, the RGB codes are often written as 8 consecutive HEX digits, see the web pages below for many examples: In a color-picker, the Teal color with R=96, G=116 and B=127, can be written in hexadecimal as #2E7471, an industry-standard notation.

4 Here is a method of converting base-10 numbers into numbers of other bases. A number N can be written as L2*B^2+L1*B^2+L0*B^0, in which L2, L1 and L0 are literals of the base. Base-two has two literals: 0,1. Base-eight (Octal) has eight literals: 0,1,2,3,4,5,6,7. Base-ten has ten literals: 0,1,2,3,4,5,6,7,8,9. Base-sixteen has sixteen literals: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. If N is divided by B, the quotient would be L2*B^1+L1*B^0 and the remainder would be L0. The operation would yield the least significant literal of the base. If the process is repeated, the next significant literal could be extract from the quotient. This process could be repeated until the quotient becomes 0. For example, the following process converts in base-ten to Octal: /8 Quotient=54534 Remainder= /8 Quotient= 6816 Remainder=6 6816/8 Quotient= 852 Remainder=0 852/8 Quotient= 106 Remainder=4 106/8 Quotient= 13 Remainder=2 13/8 Quotient= 1 Remainder=5 1/8 Quotient= 0 Remainder=1 Since the last literal is the most significant, the result is Octal. The same number converted to base-sixteen can be performed as: /16 Quotient=27267 Remainder= /16 Quotient= 1704 Remainder=3 1704/16 Quotient= 106 Remainder=8 106/16 Quotient= 6 Remainder=A 6/16 Quotient= 0 Remainder=6 The result from the above procedure is 6A834 Hex. There is a matlab utility dec2hex(436276) which yield the same result in hexadecimal. This can speed up calculation significantly for Home Work #11. The characters we use for word processing are also coded in binary. The keyboards are built consistent with the ASCII code format. ASCII is an acronym for American Standard Code for Information Interchange. It is a 7-bit code, from to (0 to 127 base-ten). The printable codes could be obtained by type ABCDE -0 in matlab. The result from that command would be , the ASCII codes for the first five capital letters. abcde -0 would return The lower case letters are +32 from the upper case letters. The value 32 is 2^5 pt in binary. Therefore, only the 2 nd bit of the 7-bit code is different. You might have noticed that many websites have a vast number of different languages on them. The character codes have expanded to 32-bit, widely known as UNICODES. Search for them and have fun!

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